Step | Hyp | Ref
| Expression |
1 | | prdsbas.p |
. . 3
⊢ 𝑃 = (𝑆Xs𝑅) |
2 | | eqid 2759 |
. . 3
⊢
(Base‘𝑆) =
(Base‘𝑆) |
3 | | prdsbas.i |
. . 3
⊢ (𝜑 → dom 𝑅 = 𝐼) |
4 | | prdsbas.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ 𝑉) |
5 | | prdsbas.r |
. . . 4
⊢ (𝜑 → 𝑅 ∈ 𝑊) |
6 | | prdsbas.b |
. . . 4
⊢ 𝐵 = (Base‘𝑃) |
7 | 1, 4, 5, 6, 3 | prdsbas 16773 |
. . 3
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
8 | | eqid 2759 |
. . . 4
⊢
(+g‘𝑃) = (+g‘𝑃) |
9 | 1, 4, 5, 6, 3, 8 | prdsplusg 16774 |
. . 3
⊢ (𝜑 → (+g‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
10 | | eqid 2759 |
. . . 4
⊢
(.r‘𝑃) = (.r‘𝑃) |
11 | 1, 4, 5, 6, 3, 10 | prdsmulr 16775 |
. . 3
⊢ (𝜑 → (.r‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
12 | | eqid 2759 |
. . . 4
⊢ (
·𝑠 ‘𝑃) = ( ·𝑠
‘𝑃) |
13 | 1, 4, 5, 6, 3, 2, 12 | prdsvsca 16776 |
. . 3
⊢ (𝜑 → (
·𝑠 ‘𝑃) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
14 | | eqidd 2760 |
. . 3
⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) |
15 | | eqidd 2760 |
. . 3
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen
∘ 𝑅))) |
16 | | eqid 2759 |
. . . 4
⊢
(le‘𝑃) =
(le‘𝑃) |
17 | 1, 4, 5, 6, 3, 16 | prdsle 16778 |
. . 3
⊢ (𝜑 → (le‘𝑃) = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
18 | | eqid 2759 |
. . . 4
⊢
(dist‘𝑃) =
(dist‘𝑃) |
19 | 1, 4, 5, 6, 3, 18 | prdsds 16780 |
. . 3
⊢ (𝜑 → (dist‘𝑃) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))) |
20 | | eqidd 2760 |
. . 3
⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
21 | | eqidd 2760 |
. . 3
⊢ (𝜑 → (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) |
22 | 1, 2, 3, 7, 9, 11,
13, 14, 15, 17, 19, 20, 21, 4, 5 | prdsval 16771 |
. 2
⊢ (𝜑 → 𝑃 = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), (+g‘𝑃)〉, 〈(.r‘ndx),
(.r‘𝑃)〉} ∪ {〈(Scalar‘ndx),
𝑆〉, 〈(
·𝑠 ‘ndx), (
·𝑠 ‘𝑃)〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑅))〉, 〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉} ∪
{〈(Hom ‘ndx), (𝑓
∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) |
23 | | prdstset.l |
. 2
⊢ 𝑂 = (TopSet‘𝑃) |
24 | | tsetid 16703 |
. 2
⊢ TopSet =
Slot (TopSet‘ndx) |
25 | | fvexd 6666 |
. 2
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) ∈ V) |
26 | | snsstp1 4699 |
. . . 4
⊢
{〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉} ⊆
{〈(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))〉, 〈(le‘ndx),
(le‘𝑃)〉,
〈(dist‘ndx), (dist‘𝑃)〉} |
27 | | ssun1 4073 |
. . . 4
⊢
{〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉,
〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉}
⊆ ({〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉,
〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉} ∪
{〈(Hom ‘ndx), (𝑓
∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}) |
28 | 26, 27 | sstri 3897 |
. . 3
⊢
{〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉} ⊆
({〈(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))〉, 〈(le‘ndx),
(le‘𝑃)〉,
〈(dist‘ndx), (dist‘𝑃)〉} ∪ {〈(Hom ‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}) |
29 | | ssun2 4074 |
. . 3
⊢
({〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉,
〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉} ∪
{〈(Hom ‘ndx), (𝑓
∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}) ⊆
(({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑃)〉, 〈(.r‘ndx),
(.r‘𝑃)〉} ∪ {〈(Scalar‘ndx),
𝑆〉, 〈(
·𝑠 ‘ndx), (
·𝑠 ‘𝑃)〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑅))〉, 〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉} ∪
{〈(Hom ‘ndx), (𝑓
∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) |
30 | 28, 29 | sstri 3897 |
. 2
⊢
{〈(TopSet‘ndx), (∏t‘(TopOpen ∘
𝑅))〉} ⊆
(({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑃)〉, 〈(.r‘ndx),
(.r‘𝑃)〉} ∪ {〈(Scalar‘ndx),
𝑆〉, 〈(
·𝑠 ‘ndx), (
·𝑠 ‘𝑃)〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑅))〉, 〈(le‘ndx), (le‘𝑃)〉, 〈(dist‘ndx),
(dist‘𝑃)〉} ∪
{〈(Hom ‘ndx), (𝑓
∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))〉, 〈(comp‘ndx), (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) |
31 | 22, 23, 24, 25, 30 | prdsvallem 16770 |
1
⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen
∘ 𝑅))) |